Ran across a question having to do with what was the interest rate if it took 12 years to double your money. Those knowiong the rule of 72 would immediately (or almost so) answer 'about 12 years". I though maybe some would be interested in how this rule was developed but that would have been to much for that section so thought I would move it to here:
First the formula: If an investment doubles in a time period t with interest rate i then
(1+i)t = 2
If we take the natural log of both sides we get
t ln(1 + i) ~ .69
Now, if i is small
ln(1+i) ~ i - 0.5 i2
Ignoring the quadratic term and converting the i into a percentage by multiplying by 100, we have
t i ~ 69
How did 69 turn into 72? My reasoning goes something like the folloiwing:
It has to do with ease of computation in the days before hand held calculators and that 'most people' like to deal in integers rather than decimals. Also, if you look at the formula up there, it looks like we would guess too low using 69 [that is, we would subtract a bit for the term 0.5 i2). What is the 'closest number with the most divisors' which is 'a bit' larger than 69? 72 is a very good candidate.
Comments?
First the formula: If an investment doubles in a time period t with interest rate i then
(1+i)t = 2
If we take the natural log of both sides we get
t ln(1 + i) ~ .69
Now, if i is small
ln(1+i) ~ i - 0.5 i2
Ignoring the quadratic term and converting the i into a percentage by multiplying by 100, we have
t i ~ 69
How did 69 turn into 72? My reasoning goes something like the folloiwing:
It has to do with ease of computation in the days before hand held calculators and that 'most people' like to deal in integers rather than decimals. Also, if you look at the formula up there, it looks like we would guess too low using 69 [that is, we would subtract a bit for the term 0.5 i2). What is the 'closest number with the most divisors' which is 'a bit' larger than 69? 72 is a very good candidate.
Comments?
